To describe geometry independently of a particular coordinate system, general relativity uses a metric tensor.

From the spacetime interval to a metric

In special relativity, the infinitesimal spacetime interval can be written schematically as

$$ds^2=\eta_{\mu\nu},dx^\mu dx^\nu,$$

where $\eta_{\mu\nu}$ is the Minkowski metric in the chosen coordinates.

General relativity replaces the constant flat-spacetime metric by a position-dependent tensor field:

$$ds^2=g_{\mu\nu}(x),dx^\mu dx^\nu.$$

The metric determines local measurements of proper time, spatial distance, angles and causal structure.

A bilinear form at each event

At each spacetime event, $g$ is a symmetric bilinear form acting on tangent vectors. In coordinates its components form the matrix $g_{\mu\nu}$.

Changing coordinates changes those components according to the tensor transformation law, while geometric quantities defined by the metric remain coordinate-independent.

Raising and lowering indices

The inverse metric $g^{\mu\nu}$ satisfies

$$g^{\mu\alpha}g_{\alpha\nu}=\delta^\mu_{\ \nu}.$$

It converts vector components to covector components and vice versa:

$$v_\mu=g_{\mu\nu}v^\nu.$$

This is not merely notation: the metric provides the specific correspondence between vectors and covectors in a given geometry.

Local flatness

At any event one can choose suitable local coordinates so that the metric takes the Minkowski form at that event and its first-order gravitational effects disappear for a freely falling observer.

That does not make the surrounding spacetime globally flat. Variation of the metric from event to event can produce curvature and tidal effects.

Geometry is encoded in a field

The metric is therefore both a geometric structure and a dynamical physical field. Once $g_{\mu\nu}$ is known, one can determine free-fall trajectories and calculate spacetime curvature. General relativity then supplies equations relating that geometry to matter and energy.